Compensation of Dynamic Friction by a Fractional Order Robust Controller

In this paper a fractional order robust control of a 2 degrees of freedom (DOF) classical mechanical system, a ball-beam system is considered. The control task has the interesting feature that only one of the DOFs of the system, i.e. the position of the ball is controlled via controlling the other axis, the tilting angle of the beam suffering from dynamic friction mathematically approximated by the LuGre model. If the internal physics of the drive system is neglected this system is a 4th order one because only the 4th time-derivative of the ball's position can directly be influenced by the torque rotationally accelerating the beam. It also has position and rotational velocity "saturation" since the gravitational acceleration limits the available acceleration of the ball both in the centripetal and in the vertical direction. This limitation is taken into account by the application of angular and angular velocity potentials keeping both values bounded. The variable structure / sliding mode controller applied is based on a standard error metrics that has to converge to zero during finite time according to a fractional order differential equation in discrete time approximation. It is shown that little reduction of the order of differentiation from 1 improves precision and robustness of the control against the measurement noises. The control is illustrated via simulation.